Fence with minimum cost
A fence is to be built to enclose a rectangular area of 300 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 12 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
Shile
1 Answer
Let $x$ and $y$ be the sides oif the rectangular region. Then
\[xy=300 \Rightarrow y=\frac{300}{x}.\]
The cost of the material is
\[C=3(x+y+x)+12 y=6x+15y=6x+15\frac{300}{x}.\]
So
\[C=6x+\frac{4500}{x}.\]
To minimize $C$ we take a derivative to find the critical points
\[C'(x)=6-\frac{4500}{x^2}=0\]
\[\Rightarrow x^2=\frac{4500}{6}=750.\]
Hence
\[x=\sqrt{750} \text{and} y=\frac{300}{\sqrt{750}},\]
are dimensions of the enclosure that is most economical to construct.
Savionf
- 1 Answer
- 414 views
- Pro Bono
Related Questions
- Volume of solid of revolution
- Derivatives
- Find the volume of the solid obtained by rotating $y=x^2$ about y-axis, between $x=1$ and $x=2$, using the shell method.
- Find value of cos(2x)
- Plot real and imaginary part, modulus, phase and imaginary plane for a CFT transform given by equation on f from -4Hz to 4Hz
- Evaluate $I(a) = \int_{0}^{\infty}\frac{e^{-ax^2}-e^{-x^2}}{x}dx $
- Finding only one real root for a function
- Find the equation of the tangent line through the function f(x)=3x$e^{5x-5} $ at the point on the curve where x=1
